The measurement of any physical property, whether the measurement is taken by a human or a machine, always includes uncertainty or error. In some cases, the act of measuring itself induces error, as taught by the famous Heisenberg uncertainty principle used frequently in quantum applications. In other cases, instruments or sensory capabilities cannot provide accurate measurements beyond a certain threshold. For example, a human being is capable of measuring the speed of a moving vehicle by observing the vehicle's motion. A human's estimate of the speed might be accurate to within 10 miles per hour, but cannot be as accurate as the measurement made by a radar gun. In this example, a vehicle's speed would be calculated by the human as, for example, 50 miles per hour plus or minus 10 miles per hour. However, a radar gun would measure the same vehicle's speed as, for example, 54 miles per hour plus or minus 1 mile per hour.
In many applications requiring measurement of physical properties, measurement error is ignored. A carpenter measuring a board with a standard tape measure can accurately measure distances to within 1/16th of an inch. In almost all applications, the carpenter can ignore the 1/16th of an inch error without ill effects. But there are many measurements that require exacting accuracy to avoid consequences. In these situations, ignoring a measurement's error can have very costly effects. More accurate measuring instruments can reduce the error to a level where it can safely be ignored, but the cost of measurement instrumentation rises as the accuracy of the instrumentation improves.
In many measurement situations, a measurement is taken to determine whether the actual value X is greater than or equal to some threshold T (X≧T). However the actual value X is elusive, and most applications must use a measured value Y that includes some measurement error e. If the error present in Y is not taken into account, several disadvantages are present:
1. If the actual value is less than the threshold, X<T, but the measurement error is large enough, the measured value will indicate Y≧T and produces an incorrect decision.
2. A corresponding disadvantage occurs if X>T but the measurement error has a large enough negative value. In this case, the measured value will indicate Y≦T and produces an incorrect decision.
3. A measurement scheme that does not account for measurement error has no way of tuning or weighting the cost of error. That is, the user of such a scheme has no way to adjust for the relative cost of the two types of error (i.e. a false positive and a false negative). For example, in the case of a hot water boiler, if the measured value of a boiler's internal pressure is greater than a set threshold, a relief valve will open to avoid a catastrophic overpressure explosion. In the boiler example, the consequences of a false negative (measured pressure is less than threshold because of negative error, but actual pressure is above threshold so the boiler explodes) are much higher than the consequences of a false positive (measured pressure is greater than threshold because of positive error, but actual pressure is below threshold so steam is vented prematurely to relieve pressure).
4. A measurement scheme that does not account for measurement error ignores the a priori probability of the actual value being equal to or greater than a threshold. For example, in the case of a very high threshold, the actual value of a property may be less than a threshold in almost all cases, so that any measured values of that property exceeding the threshold are almost certainly the result of measurement error. Accordingly, this scheme makes a wrong decision with high probability.
Some prior art methods add or subtract the standard deviation (a) of the uncertainty of a measured value from a threshold to which the measured value is compared. Using the boiler example, this scheme would test for Y≧T−τ, rather than Y≧T, where Y is the measured pressure value and T is the threshold. This scheme accepts more false positives in return for fewer false negatives. In situations where a false positive is more costly, the user can test for Y≧T+τ. This scheme avoids disadvantages 1 and 2 above by automatically adjusting as the error variance becomes larger or smaller, and partly avoids disadvantage 3 in that the user can weight one type of error more heavily than the other. However, it does not allow a user to specify how much to weight one type of error over another. This scheme does not avoid disadvantage 4 as there is no provision accounting for the distribution of actual values compared to a threshold.
Ignoring the error of a measured property can be costly in other ways. In a manufacturing setting, measurement errors are often accounted for by setting a lower threshold for discarding a product. For example, if a manufactured resistor must have a value of 100 ohms and measurements can only return a value of plus or minus 5 ohms, in any situation where the actual value of the resistor must be greater than or equal to 100 ohms, all resistors with measured values of 105 ohms or less must be discarded. In this example and in many present scenarios, buying a measurement system of greater accuracy is a large capital expense. Therefore, methods that would better account for measurement error and the associated costs of the error are needed.